Diffraction Tomography in Dispersive Backgrounds

1. Diffraction Tomography in Dispersive Backgrounds Tony Devaney Dept. Elec. And Computer Engineering Northeastern University Boston, MA 02115 Email: tonydev2@aol.com A.J. Devaney, “Linearized inverse scattering in attenuating media,” Inverse Problems3 (1987) 389-397 Other approaches discussed in: • A. Schatzberg and A.J.D., ``Super-resolution in diffraction tomography, Inverse Problems 8 (1992) 149-164 • K. Ladas and A.J.D., ``Iterative methods in geophysical diffraction tomography, Inverse Problems 8 (1992) 119-132 • R. Deming and A.J.D., ``Diffraction tomography for multi-monostatic gpr, Inverse Problems 13 (1997) 29-45

2. Experimental Configuration n() O(r,) s0 s Generalized Projection-Slice Theorem E. Wolf, Principles and development of diffraction tomography, Trends in Optics, Anna Consortini, ed. [Academic Press, San Diego, 1996] 83-110

3. Forward scatter data Back scatter data z Born Inverse Scattering k=real valued Ewald Spheres k 2k Ewald Sphere Limiting Ewald Sphere

4. Born Inversion for Fixed Frequency Problem: How to generate inversion from Fourier data on spherical surfaces Inversion Algorithms: Fourier interpolation (classical X-ray crystallography) Filtered backpropagation (diffraction tomography) A.J.D. Opts Letts, 7, p.111 (1982) Filtering of data followed by backpropagation: FilteredBackpropagationAlgorithm Fourier based methods fail if k is complex: Need new theory

5. Pulse Propagation in a Dispersive Background n() O(r,) s0 s

6. Fourier Transformed Scattered Field Close in u.h.p. Choose a complex frequency 0 such that k(0 ) is real valued There is no reason a priori to dismiss this possibility, butwill it work? Roots of dispersion relationship with real k are in l.h.p.

7. Simple Conducting Medium Complex in l.h.p. Real valued Im  Complex  plane Branch point X <0 X Desired frequency 0 Re  Will not be able to close in u.h.p.: can only drop contour to branch points

8. Lorentz Model b2=20x1032 0=16x1016  =.28x1016 Real n Imag n K.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics [Springer-Verlag, 1994, New York]

9. Lorentz Medium Im  Complex  plane Re  <0 - Desired frequency 0 X - + x x Poles of n() Branch Cuts Roots of dispersion relationship must lie above branch points Im 0>-

10. Contour Plot of Re ik() Im  Re  Real k Branch point

11. Mesh Plot of Re ik()

12. Exciting the Plane Wave n() s0 O(r,) Close in l.h.p. Non-attenuating mode of medium

13. The Complete Pulse Im  Complex  plane Re  -0 0 X X Branch Cuts Precursors Can the non-attenuating plane wave be excited; i.e., is it dominated by the precursors?

14. Asymptotic Analysis K.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics [Springer-Verlag, 1994, New York] Im  Steepest Descent Contour Complex  plane Saddle point X Re  -0 0 X X X X X Saddle point X Saddle point Plane wave excited Plane wave not excited

15. Summary and Questions • Have reviewed one possible approach to inversion in dispersive backgrounds • Method is based on computing the temporal Fourier transform of pulsed data • at complex frequencies for which the wavenumber of the background is real • Method will not work for simple conducting media but appears feasible for • Lorentz media • The idea behind the approach suggests that it may be possible to excite • non-decaying, plane wave pulses using complex frequencies • Asymptotic analysis is required to determine the feasibility of the theory